Hidden Topological Angles in Path Integrals

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PRL 115, 041601 (2015)

Hidden Topological Angles in Path Integrals Alireza Behtash,* Tin Sulejmanpasic,† Thomas Schäfer,‡ and Mithat Ünsal§ Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA (Received 13 March 2015; published 24 July 2015) We demonstrate the existence of hidden topological angles (HTAs) in a large class of quantum field theories and quantum mechanical systems. HTAs are distinct from theta parameters in the Lagrangian. They arise as invariant angles associated with saddle points of the complexified path integral and their descent manifolds (Lefschetz thimbles). Physical effects of HTAs become most transparent upon analytic continuation in nf to a noninteger number of flavors, reducing in the integer nf limit to a Z2 valued phase difference between dominant saddles. In N ¼ 1 super Yang-Mills theory we demonstrate the microscopic mechanism for the vanishing of the gluon condensate. The same effect leads to an anomalously small condensate in a QCD-like SUðNÞ gauge theory with fermions in the two-index representation. The basic phenomenon is that, contrary to folklore, the gluon condensate can receive both positive and negative contributions in a semiclassical expansion. In quantum mechanics, a HTA leads to a difference in semiclassical expansion of integer and half-integer spin particles. DOI: 10.1103/PhysRevLett.115.041601

PACS numbers: 11.15.-q, 11.15.Kc, 11.15.Tk

Introduction.—Providing a nonperturbative continuum definition of the path integral in quantum field theory is a challenging but important problem [1]. There is growing evidence that, if an ordinary integral or a path integral admits a Lefschetz-thimble decomposition [2,3] or resurgent transseries expansion [4–12] then either of these methods gives this long-sought nonperturbative definition. If this is indeed the case, then we expect that these new methods will provide new and deep insight into quantum field theory and quantum mechanics formulated in terms of path integrals. In this article we introduce a new phenomenon of this kind, the appearance of hidden topological angles (HTAs). The main prescription associated with the Lefschetzthimble decomposition or the resurgent expansion is the following: Even if an ordinary integral or a path integral is formulated over real fields, the natural space that the critical points (saddles) ρσ live in is the complexification of the original space of fields. However, the dimension of the critical point cycles J σ is that of the original space, or half that of the complexified field space. For example, for an ordinary integral over N-dimensional real space, P this N N N N procedure is R → C → Σ , where Σ ¼ σ nσ J σ and dimR ðJ σ Þ ¼ N. For N ¼ 1, this is the well-known steepest descent (stationary phase) approximation. To each critical point ρσ of the complexified action one attributes an action, with real and imaginary parts, and with “weight” e−Sσ . The imaginary part of the action, ImSσ is an invariant angle associated with the critical point ρσ and its descent manifold J σ . If there are critical points with the identical real part of the action ReSσ , but different imaginary parts ImSσ , then there may be subtle effects. Indeed, Witten recently studied in Ref. [3] the analytic continuation of Chern-Simons theory to noninteger values 0031-9007=15=115(4)=041601(5)

of the coupling k, finding subtle cancellations among dominant saddle field configurations in the integer k limit, so that the subdominant saddle gives the main physical contribution. In this work, we show that the effect observed by Witten is not an exotic phenomenon, but that it is possibly quite ubiquitous, and that it is responsible for a variety of interesting physical effects in quantum field theories and quantum mechanics, in which the analytic continuation is now noninteger “coupling” nf , which is the number of fermionic flavors for integer values. We also show that the effect is more nontrivial than a simple cancellation between dominant saddles. Indeed, the effect depends on the observable, and the dominant saddles may cancel in certain observables, but contribute to others. In a field theory with a topological Θ angle in the Lagrangian, subtle effects may arise at certain values of the Θ angle [13–16]. For example, at Θ ¼ π in SUð2Þ gauge theory there is a cancellation of leading order saddle contribution to the mass gap [14]. In this work we study a more exotic phenomenon, which is due to a hidden topological angle not explicitly present in the Lagrangian. We define a HTA as the phase associated with a saddle point in the complexified field space. HTA may depend on the number of fermionic flavors nf or spin of a particle S, and may be interpreted as topology of a saddle in the complexified field space. Below, we will provide examples of this phenomenon in N ¼ 1 super Yang-Mills (SYM) theory, certain QCD-like theories, and the quantum mechanics of a particle with spin. We also note that a HTA is different from the discrete theta angles discussed recently [17], which comes about as one changes the global gauge group. In contrast, HTAs are present for any gauge group. In the examples discussed below we find that for

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integer values of the number of fermions nf , there is a Z2 hidden topological structure. A prototype in ordinary integration.—An elementary example that provides some intuition for field theory is the following. Consider the analytic continuation of the Bessel function to noninteger order, and describe the contour that appears in the integral representation in terms of Lefschetz thimbles. In one complex dimension the Lefschetz thimble is defined as a stationary phase manifold: Im½SðwÞ − Sðwn Þ ¼ 0, where wn is aR critical point on the contour. The integral is Iðk; λÞ ¼ Cw dwe2λ sinhðwÞþkw for complex k, λ. In a certain regime of the analytic continuation, discussed in Ref. [3], the integral can be expressed in terms of three cycles, J i ; i ¼ 1; 2; 3 associated with saddles ρi , so that Cw ¼ J 1 þ J 2 þ J 3 ; see Fig. 1. The sum of the three thimble contribution gives ð1 þ e2πi½kþð1=2Þ Þe−S1 þ e−S2 ;

phase, Im½SðwÞ − Sðwn Þ ¼ 0, is only satisfactory for a one-dimensional integral (it provides one real condition on a one-complex dimensional space). In n complex dimensions, where n ¼ ∞ corresponds to field theory or quantum mechanics, this condition defines a co-dimension one (real dimension 2n − 1 space), which is not the desired n real dimensional space. Instead, one needs n real conditions to define the thimble. Guided by these observations, Witten used complex gradient flow equations, the PicardLefschetz equations, to describe the Lefschetz thimbles. In a theory with a field φ and action SðφÞ, this amounts to ∂φ δS¯ ¼− ; ∂τ δφ¯

FIG. 1 (color online). The blue areas show “good regions” in which the integrand falls sufficiently rapidly at infinity to guarantee convergence. The red dots give the locations of the saddle points, and the blue contours are the Lefschetz thimbles. If the boundary of integration is (−∞; −∞ þ 2πi), then the Lefschetz decomposition is J 1 þ J 2 þ J 3 . Here, ρ1 and ρ3 are equally dominant saddles over ρ2, but there is an overall phase difference between the dominant saddles leading to a subtle cancellation for integer k.

ð2Þ

where τ is the flow time. Using Eq. (2) and the chain rule,   ∂Im½S 1 δS ∂φ δS¯ ∂ φ¯ ¼ − ¼ 0; ∂τ 2i δφ ∂τ δφ¯ ∂τ

ð1Þ

where je−S1 j ¼ je−S3 j ≫ e−S2 ; i.e., ρ1 and ρ3 are dominant over ρ2. However, they have a relative phase, and the contribution of these two dominant saddles cancels each other exactly for integer k. The mechanism described above is an intuitive example of a mechanism operative in ChernSimons theory by using analytic continuation, providing confidence for the utility of the idea of analytic continuation of path integrals. Other examples are discussed in Refs. [18–20]. We will perform a similar analytic continuation in nf , the number of fermion flavors in the theory. Picard-Lefschetz equation and invariant angles.—The definition of the Lefschetz thimble based on the stationary

∂ φ¯ δS ¼− ; ∂τ δφ

ð3Þ

meaning that Im½SðϕÞ ¼ Im½Sðϕn Þ is invariant under the flow. In a quantum field theory (QFT), or in quantum mechanics (QM), in which a semiclassical saddle proliferates (an example is the instanton gas), Im½Sðϕn Þ will appear as a genuinely new phase in the effective field theory. This is the HTA phenomenon. The integration in the complexified field space is infinite dimensional. In the background of nonperturbative saddles, this space usually factorizes into finite dimensional zero and quasizero mode directions and infinite dimensional Gaussian modes. The HTA can be calculated by an exact integration over the complexified finite dimensional quasizero mode directions in the field space, dictated by the finite dimensional version of the Picard-Lefschetz theory. Hidden topological angle in 4d N ¼ 1 SYM theory.— Consider N ¼ 1 SYM theory on R3 × S1L , where S1L is a circle with period L. We use supersymmetry preserving boundary conditions and take the small L limit in order to be able to use semiclassical methods. According to the trace anomaly relation, the gluon condensate hð1=NÞtrF2μν i determines the vacuum energy: E vac ¼ hΩjT 00 jΩi ¼ 1 1 3 2 4 hΩjT μμ jΩi ¼ 4 ½βðgÞ=g htrFμν i. This implies that the gluon condensate can serve as an order parameter for supersymmetry breaking. The vacuum energy density, and hence the condensate, vanishes to all orders in perturbation theory in supersymmetric theories. Since supersymmetry is known to be unbroken, the gluon condensate must be zero nonperturbatively as well. In the semiclassical limit this result appears mysterious, because all contributions appear to be positive. The reason is that in Euclidean space the fermion determinant is positive definite, and trF2μν is also positive definite. This implies that the gluon condensate is the average of a positive observable with respect to a positive measure [21]. Then, how does the vanishing of the htrF2μν i take place from a semiclassical point of view?

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We address this question in the regime of small, but finite radii on R3 × S1L . To do so, recall the Euclidean realization of the vacuum of the theory on small R3 × S1L , depicted in Fig. 2 for the center-symmetric point of the Wilson line on the Coulomb branch. The vacuum is, primarily, a dilute gas of semiclassical one and two events: monopole-instantons [22–25] and bions [5,26–28]. These are (i) monopoleinstantons, Mi ¼ e−S0 ðαi · λÞ2 , (ii) magnetic bions, Bij ¼ ½Mi Mj  ¼ e−2S0 …, (iii) neutral bions, Bii ¼ ½Mi Mi  ¼ e−2S0 þiπ …, where αi ; i ¼ 1; …; N are simple roots complemented with the affine root αN , and Bij and Bii are nonvanishing ∀Aˆ ij < 0, and ∀Aˆ ii > 0 entries of the extended Cartan matrix, respectively. The monopole action is S0 ¼ ð8π 2 =g2 NÞ. For small L the coupling is small, the action is large, and fluctuations are suppressed. The 2N fermion zero modes of the 4d instanton are distributed uniformly as (2; 2; …; 2) to monopoles Mi . At leading order Oðe−S0 Þ in the semiclassical limit, each monopole-instanton has two fermion zero modes and therefore they do not contribute to the gluon condensate. Two-defects do contribute to the gluon condensate. For the sake of making the analogy with the toy example (1) explicit, let us consider analytic continuation away from nf ¼ 1. The density of both types of 2-defects is the same, of order Oðe−2S0 Þ. However, there is an extra ð4nf − 3Þπ phase (invariant angle) associated with the neutral bion saddle or thimble: ArgðJ Bii Þ ¼ ArgðJ Bij Þ þ ð4nf − 3Þπ:

ð4Þ

Consequently, in contrast to the folklore regarding the positivity of the gluon condensate, the contributions of the two types of 2-defects to the gluon condensate cancel

L4 hð1=NÞtrF2μν i ¼ 0 × nMi þ ðnBij þ eið4nf −3Þπ nBii Þ ¼ 0 ð5Þ at a physical integer value of the parameter nf ¼ 1 similar to Eq. (1). This is the microscopic mechanism for the vanishing of the gluon condensate as well as the vacuum energy in N ¼ 1 SYM theory. The two contributing bion thimbles are charged oppositely under the ZHTS 2 , and cancel each other out. The difference with respect to the toy integral and the cancellation in analytically continued Chern-Simons theory is the fact that this cancellation is observable dependent. In fact, the combination of the neutral and magnetic bions, despite giving a vanishing contribution to gluon condensate, is responsible for the formation of a mass gap. To see this, consider the effective Lagrangian for the low energy bosonic modes. As an example, we will use SUð2Þ gauge theory. Let ϕ denote the fluctuation of the Wilson line around the center symmetric minimum and σ denote the dual photon. The bosonic potential induced by 2-defects is Ref. [29] Vðσ; ϕÞ ¼ −ðB12 þ B21 þ B11 þ B22 Þ ∼ e−2S0 ð− cos 2σ − eiπ cosh 2ϕÞ:

We observe that the factor eiπ responsible for the vanishing hð1=NÞtrF2μν i is also responsible for the (positive and unsuppressed) mass gap of the ϕ fluctuations and stabilizes center symmetry. The HTA explains both the vanishing of the gluon condensate and the nontachyonic nature of fluctuations of the Polyakov line. For a phenomenological interpretation of the negative sign see [30]. It is also not particular to supersymmetric theory, as we discuss next. QCD[AS/S].—In a typical confining asymptotically free SUðNÞ gauge theory, the “natural” scaling of the (properly normalized) gluon condensate is OðN 0 Þ: hð1=NÞtrF2μν i ∝ N 0 Λ4 . It is natural to expect that the vanishing of the gluon condensate is special to the supersymmetric theory. This is not the case. There exists an exact large-N orientifold/ orbifold equivalence between N ¼ 1 SYM theory and QCD[AS/S] [31], proven in Ref. [32]. Here, AS(S) refers to fermions in antisymmetric(symmetric) two-index representations. The large-N equivalence implies that the gluon condensate in QCD[AS/S] is zero at leading order in the 1=N expansion, and must scale as [33] 

FIG. 2 (color online). A snapshot of the Euclidean vacuum of N ¼ 1 SYM theory on small R3 × S1L . Both neutral and magnetic bions carry action 2S0 , but their contribution to gluon condensate cancels exactly because of the presence of a HTA, a π-phase difference between the two saddles.

ð6Þ

1 trF2μν N

QCDðAS=SÞ

¼

1 4 Λ; N

ð7Þ

in sharp contrast with the natural value. This result is counterintuitive, but it is a rigorous consequence of the large N equivalence. However, as in the supersymmetric case, there is no known semiclassical explanation. Again, we can understand the result based on the presence of HTAs. To achieve this, we use the framework

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of deformed Yang-Mills theory, and add AS representation fermions [a similar analysis holds for QCD(S)]. In QCD (AS), the Atiyah-Singer index theorem implies that the number of fermion zero modes of a 4d instanton is 2N − 4. There are N types of monopole-instantons, with the number of fermion zero modes distributed as (2; 2; …; 2; 0; 0) in a center-symmetric background. The difference with respect to N ¼ 1 SYM theory is that 2 out of N monopoleinstantons do not possess Fermi zero modes. Therefore, at leading order, Oðe−S0 Þ, in the semiclassical expansion, N − 2 monopoles do not contribute to the gluon condensate and only two do, giving a positive contribution proportional to 1=N. At second order in the semiclassical expansion Oðe−2S0 Þ there are magnetic and neutral bions that can contribute to gluon condensate. Their contribution cancels at leading order in N, analogous to SYM theory, leading to Eq. (7). In the older literature on QCD [21,34,35], it was assumed that in the semiclassical limit gluon condensate is proportional to the instanton density. This was based on the rationale that a single Rinstanton contributes a finite and positive amount, ð1=2g2 Þ trF2μν ¼ ð8π 2 =g2 Þ, and that the condensate can be attributed to 4d instantons with a positive weight hð1=NÞtrF2μν i ∝ nI . In the calculable small S1L regime, we see that this is incorrect in at least two ways: (i) Instantons are subleading, i.e.,Oðe−N Þ, (ii) the weight of the saddles can be both positive (decreasing energy) and negative (increasing energy). Our work is the first example in which a contribution to the condensate has a negative component. Quantum mechanics.—In order to show the generality of hidden topological angles, we also consider the quantum mechanics of a particle with position xðtÞ and internal spin ð12Þnf . The Euclidean Lagrangian is that of a bosonic field xðtÞ coupled to nf fermionic fields ψ i :   1 2 1 0 2 00 x_ þ ðW Þ þ ðψ¯ i ψ_ i þ W ψ¯ i ψ i Þ : LE ¼ 2 2

ð8Þ

For nf ¼ 1, this theory is supersymmetric [36]. If we choose WðxÞ to be a periodic function, for example, WðxÞ ¼ cos x, we may identify x ¼ x þ 2π as the same physical point, corresponding to a particle on a circle, rather than in an infinite lattice. The system contains two types of instantons, I 1 ∶ ½0 → π;

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I 2 ∶ ½0 → −π:

ð9Þ

Here, I 2 is an instanton (not an anti-instanton), because it satisfies the same BPS equation (or gradient flow equation, if W is viewed as a Morse function) as I 1 . Because of spin, instantons do not contribute to the vacuum energy. A nonvanishing contribution arises from correlated two-events. This parallels the 4d field theory on R3 × S1 , and provides a simple system in which the effect

of the HTA is not contaminated by first order instanton effects. Following Refs. [37,38], we find that the amplitudes of the two-events are given by ½I 1 I¯ 1  ¼ ½I 2 I¯ 2  ∝ eiπnf e−2SI , and ½I 1 I¯ 2  ¼ ½I 2 I¯ 1  ∝ e−2SI . Because of the difference in the invariant angles between the twosaddles or thimbles we find ArgðJ ½I 1 I¯ 1  Þ ¼ ArgðJ ½I 1 I¯ 2  Þ þ nf π:

ð10Þ

The nonperturbative contribution to the ground state energy takes the form iπnf −2SI ΔEnp Þe : 0 ¼ ð−2 − 2e

ð11Þ

While the ½I 1 I¯ 2  molecule behaves in the expected manner and decreases the ground state energy, the ½I 1 I¯ 1  molecule is sensitive to the HTA governed by the spin and increases the ground state energy for odd nf (half-integer spin) while decreasing it for even nf (integer spin). In the case of nf ¼ 1, Eq. (10) is the microscopic reason for the nonperturbative vanishing of the ground state energy. We note that this system, despite having Witten index zero [39], I W ¼ tr½ð−1ÞF  ¼ 0, has unbroken supersymmetry, and two supersymmetric ground states. There are two additional interesting features of this system. The first is related to the fact that one can introduce R a topological angle into the Lagrangian, iðΘ=2πÞ x_ dτ, and, unlike the case of supersymmetric gauge theory, the Θ angle is physical, and alters the spectrum of the theory. Since I W ¼ 0, the supersymmetry of this system is fragile. The vacuum energy can be written as iπnf Þe−2SI ; ΔEnp 0 ¼ ð−2 cos Θ − 2e

ð12Þ

which, for the supersymmetric theory (nf ¼ 1), takes the form −2SI ΔEnp ¼ 4sin2 0 ¼ ð−2 cos Θ þ 2Þe

Θ −2SI e ≥ 0; ð13Þ 2

meaning supersymmetry is dynamically broken for Θ ≠ 0. Note that the energy remains positive semidefinite, which is a consequence of the supersymmetry of the Hamiltonian. The physical reason for E > 0 in the case Θ ≠ 0 is that the Θ angle is equivalent to feeding momentum into the system. Because of supersymmetry, a bosonic or fermionic ground state pair is lifted simultaneously by the insertion of momentum, leading to a nonvanishing ground state energy. The second unusual feature is that the theory has Witten index I W ¼ 0 for any value of Θ, but that the reason for I W ¼ 0 differs in the two cases. For Θ ¼ 0, we get I W ¼ 1 − 1 ¼ 0, where the two contributions arise from the bosonic or fermionic sectors of the Hilbert space, and supersymmetry is unbroken. In the second case, Θ ≠ 0, we get I W ¼ 0 − 0 ¼ 0 and supersymmetry is broken.

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One may speculate that the invariant angles are related to Berry phases [40], realized in terms of Euclidean saddles, at least in the case of quantum mechanics. Since ð12Þnf ¼ ⊕multðSÞS, where multðSÞ is the multiplicity, we S

can rewrite the path integral over the Grassmann variables P as spin path integrals [41] Z ¼ S multðSÞZðSÞ , where Z R ðSÞ Z ¼ DxDðcos θÞDϕe−LE þiðΘ=2πÞ x_ dτ ; 1 _ LE ¼ ð_x2 þ ðW 0 Þ2 Þ þ SW 00 cos θ þ iSð1 − cos θÞϕ; 2 ð14Þ where ðθ; ϕÞ ∈ S2 parametrize the Bloch sphere. There are two spin dependent interactions, a “magnetic field” W 00 -spin coupling, and the Wess-Zumino term, or Berry phase action. In this language Eq. (10) should be replaced by ArgðJ ½I 1 I¯ 1  Þ ¼ ArgðJ ½I 1 I¯ 2  Þ þ 2Sπ. This distinguishes half-integer and integer spin particles, similar to antiferromagnets in one-spatial dimension [42], leading to qualitative differences. Conclusion.—We have provided several examples of Z2 hidden topological angles, associated with the saddle point manifolds that appear in the complexified path integral, and have shown that these angles lead to crucial physical effects. We anticipate that HTAs will have crucial impact on the semiclassical analysis of many interesting quantum field theories and quantum mechanical systems. Our examples also show that in an attempt to perform lattice simulations using Lefschetz thimbles, e.g., Refs. [43–46], all thimbles whose multipliers are nonzero must be carefully summed over to correctly capture the dynamics of the theory. We thank G. Dunne, G. ’t Hooft, P. Argyres, and E. Witten for useful comments and discussions. We acknowledge support from DOE Grants No. DE-FG02-03ER41260 and No. DE-SC0013036.

*

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